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Joint Density Poisson arrival
Let \(T_1\) and \(T_5\) be the times of the first and fifth arrivals in a Poisson arrival prices with rate \(\lambda\). Find the joint distribution of \(T_1\) and \(T_5\) .
Telephone Calls throughout the Week
Telephone calls come in to a customer service hotline. The number of calls that arrive within a certain time frame follows a Poisson distribution. The average number of calls per hour depends on the day of the week. During the week (Monday through Friday) the hotline receives an average of 10 calls per hour. Over the weekend (Saturday and Sunday) the hotline receives and average of 5 calls per hour. The hotline operates for 8 hours each day of the week. (The number of calls on one day is independent of the numbers of calls on other days.)
- What is the probability that the center receives more than 500 calls in 1 week?
- Each person who calls the center has a 20% chance of getting a refund (independent of other callers). Find the probability that 10 or fewer people get a refund on Tuesday.
- One day of the week is chosen uniformly at random. On this day, a representative at the call center reports that 60 people called in. Based on that information, what is the probability that the day was a weekend day (either Saturday or Sunday)?
Population
A population contains \(X_n\) individuals at time \(n=0,1,2,\dots\) . Suppose that \(X_0\) is distributed as \(\mathrm{Poisson}(\mu)\). Between time \(n\) and \(n+1\) each of the \(X_n\) individuals dies with probability \(p\) independent of the others. The population at time \(n+1\) is comprised of the survivors together with a random number of new immigrants who arrive independently in numbers distributed according to \(\mathrm{Poisson}(\mu)\).
- What is the distribution of \(X_n\) ?
- What happens to this distribution as \(n \rightarrow \infty\) ? Your answer should depended on \(p\) and \(\mu\). In particular, what is \( \mathbf{E} X_n\) as \(n \rightarrow \infty\) ?
[Pittman [236, #18]
Poisson Thinning
Let \(N(t)\) be a Poisson process with intensity λ. For each occurrence, we flip a coin: if heads comes up we label the occurrence green, if tails comes up we label it red. The coin flips are independent and \(p\) is the probability to see heads.
- Show that the green occurrence form a Poisson process with intensity λp.
- Connect this with example 2.2.5 from Meester.
- We claim that the red occurrences on the one hand, and the green occurrences on the other hand form independent Poisson processes. Can you formulate this formally, and prove it , using Example 2.2.5 from Meester once more?
[Meester ex. 7.5.7]
Closest Point
Consider a Poisson random scatter of points in a plane with mean intensity \(\lambda\) per unit area. Let \(R\) be the distance from zero to the closest point of the scatter.
- Find a formula for the c.d.f. and the density of \(R\) and sketch their graphs.
- Show that \(\sqrt{2 \lambda \pi} R\) has the Rayleigh distribution.
- Find the mean and mode of \(R\).
[pitman p 389, # 21]
Joint Density of Arrival Times
Let \(T_1 < T_2<\cdots\) be the arrival times in a Poisson arrival process with rate \(\lambda\). What is the joint distribution of \((T_1,T_2,T_5)\) ?
Raindrops are falling
Raindrops are falling at an average rate of 30 drops per square inch per minute.
- What is the chance that a particular square inch is not hit by any drops during a given 10-second period ?
- If one draws a circle of radius 2 inches on the ground, what is the chance that 4 or more drops hits inside the circle over a two-minute period?
- If each drop is a big drop with probability 2/3 and a small drop with probability 1/3, independent of the other drops, what is the chance that during 10 seconds a particular square inch gets hit by precisely four big drops and five small ones?
[Pitman p. 236, #17, Modified by Mattingly]
Defective Machines
Suppose that the probability that an item produced by a certain machine will be defective is 0.12.
- Find the probability (exactly) that a sample of 10 items will contain at most 1 defective item.
- Use the Poisson to approximate the preceding probability. Compare your two answers.
[Inspired Ross, p. 151, example 7b ]
Boxes without toys
A cereal company advertises a prize in every box of its cereal. In fact, only about 95% of the boxes have a prize in them. If a family buys one box of this cereal every week for a year, estimate the chance that they will collect more than 45 prizes. What assumptions are you making ?
[Pitman p122, # 9]
Joint arrival times
Let \(T_1\) and \(T_5\) be the times of the first and fifth arrival in a Poisson process with rate \(\lambda\). Find joint density of \(T_1\) and \(T_5\).
[Pitman p355 #12]
Simple Poisson Calcuations
Let \(X\) have Poisson\((\lambda)\) distribution. Calculate:
- \(\mathbf{E}(3 X +5)\)
- \(\mathbf{Var}(3X +5)\)
- \(\mathbf{E}\big[\frac1{1+X} \big]\)
Mixing Poisson Random Variables 1
Assume that \(X\), \(Y\), and \(Z\) are independent Poisson random variables, each with mean 1. Find
- \(\mathbf{P}(X+Y = 4) \)
- \(\mathbf{E}[(X+Y)^2]\)
- \(\mathbf{P}(X+Y + Z= 4) \)
Random Errors in a Book
A book has 200 pages. The number of mistakes on each page is a Poisson random variable with mean 0.01, and is independent of the number of mistakes on all other pages.
- What is the expected number of pages with no mistakes ? What is the variance of the number of pages with no mistakes ?
- A person proofreading the book finds a given mistake with probability 0.9 . What is the expected number of pages where this person will find a mistake ?
- What, approximately, is the probability that the book has two or more pages with mistakes ?
[Pitman p235, #15]
Sums of Poisson
Agambler bets ten times on events of probability \(1/10\), then twenty times on events with probability \(1/20\), then thirty times on events with probability \(1/30\), then forty times on events with probability \(1/40\). Assuming the vents are independent, what is the approximate distribution of the number of times the gambler wins ? (use Poisson approx. of binomial)
[Pitman 2.5, pg 227]
Conditional Poisson
The following is a hierarchical model.
- \(\lambda \sim Uniform[1,2]\)
- \(Y \mid \lambda \sim \mbox{Poisson}(\lambda)\)
What is \(\mathbf{E}(Y)\) ?
Mixture of Poisson
The following is a mixture model. The following experiment is used to draw a random variable \(Y\). With probability \(p\) draw from a Poisson distribution with parameter \(\lambda = 1\) so with probability \(1-p\) you are drawing from a Poisson distribution with parameter \(\lambda =2 \).
What is \(\mathbf{E}(Y)\) ?
Approximation: Rare vs Typical
Let \(S\) be the number of successes in 25 independent trials with probability \(\frac1{10}\) of success on each trial. Let \(m\) be the most likely value of S.
- find \(m\)
- find the probability that \(\mathbf{P}(S=m)\) correct to 3 decimal places.
- what is the normal approximation to \(\mathbf{P}(S=m)\) ?
- what is the Poisson approximation to \(\mathbf{P}(S=m)\) ?
- repeat the first part of the question with the number of trial equal to 2500 rather than 25. Would the normal or Poisson approximation give a better approximation in this case ?
- repeat the first part of the question with the number of trial equal to 2500 rather than 25 and the probability of success as \(\frac1{1000}\) rather that \(\frac1{10}\) . Would the normal or Poisson approximation give a better approximation in this case ?
[Pitman p122 # 7]
